AI 中文总结
针对具有孤立临界点的全纯芽，研究其负梯度流在原点的康利指数，证明该指数是米尔诺纤维的悬置，进而建立米尔诺数与同调康利指数秩的关系，证明过程利用了全纯莫尔斯化及康利指数的和性质。
AI 中文摘要
设$f:(\C^n,0)\to(\C,0)$是在原点具有孤立临界点的全纯芽，$\varphi_f$是$\re f$关于黎曼度量的负梯度流。我们证明原点是$\varphi_f$的孤立不变集，其康利指数是米尔诺纤维的悬置，即\\[ h(\varphi_f,\{0\})\\;\simeq\\;\Sigma F_f\\;\simeq\\;\bigvee_{\mu(f)}S^n, \\]因此$\mu(f)=\rank \CH_n(\varphi_f,\{0\};\Z)$：米尔诺数是该流在$n$次的同调康利指数的秩。证明过程利用了延拓至$f$的全纯莫尔斯化。由于$\im f$是该流的首次积分，在将目标旋转一个避开有限多个值的角度后，所得的$\mu(f)$个非退化平衡点之间不存在连接轨道，且康利指数的和性质适用。随后麦科德定理得到$-\nabla\re f$的庞加莱-霍普夫指数。
英文摘要
Let $f:(\C^n,0)\to(\C,0)$ be a holomorphic germ with an isolated critical point at the origin, and let $φ_f$ be the negative gradient flow of $\re f$ with respect to a Riemannian metric. We show that the origin is an isolated invariant set of $φ_f$ and that its Conley index is the suspension of the Milnor fiber, \[ h(φ_f,\{0\})\;\simeq\;ΣF_f\;\simeq\;\bigvee_{μ(f)}S^n , \] so that $μ(f)=\rank \CH_n(φ_f,\{0\};\Z)$: the Milnor number is the rank of the homological Conley index of the flow in degree $n$. The proof uses continuation to a holomorphic morsification of $f$. Since $\im f$ is a first integral of the flow, after a rotation of the target through an angle avoiding finitely many values there are no connecting orbits between the $μ(f)$ nondegenerate rest points that result, and the sum property of the Conley index applies. McCord's theorem then recovers the Poincaré--Hopf index of $-\nabla\re f$.
